Gottfried Leibniz · Mathematics
How Leibniz developed the base-two number system - every number written with only 0 and 1 - finding in it a metaphysical image of creation and laying, unknowingly, the arithmetical foundation of the digital age.
We ordinarily count in base ten: ten digits, 0 through 9, and when we run out we start a new column worth ten times as much. The number 237 means two hundreds, three tens, and seven ones - each column ten times the one to its right. There is nothing sacred about ten; we use it because we have ten fingers. Leibniz asked: what if we used the smallest possible base, just two digits, 0 and 1? Then each column would be worth twice the one to its right - ones, twos, fours, eights, sixteens, and so on, the powers of two.
In this binary system, the number we call thirteen is written 1101, meaning one eight, one four, no twos, and one one: 8 + 4 + 0 + 1 = 13. The number five is 101 (one four, no twos, one one). Counting goes 1, 10, 11, 100, 101, 110, 111, 1000 - for one, two, three, four, five, six, seven, eight. Every whole number, no matter how large, can be written with nothing but zeros and ones. Leibniz worked out the full arithmetic of this system - how to add, subtract, multiply, and divide in base two - and found the rules astonishingly simple, far simpler than the decimal multiplication tables children labour to memorise. To add in binary you need only know that 1 + 1 = 10 (carry the one); to multiply, you only ever multiply by 0 or 1. The whole of arithmetic, reduced to its barest possible elements.
For Leibniz, binary was not merely a calculating convenience; it was charged with metaphysical and even religious meaning. He was struck that every number could be generated from just two symbols - 1 and 0, which he read as something and nothing, being and the void. Here, it seemed to him, was an arithmetical image of the deepest truth of his theology: that God created the entire universe out of nothing, that all of being unfolds from the divine unity (the 1) and the void (the 0). As early as 1697 he had a commemorative medal designed, depicting the creation of the world with the motto that one suffices to draw all things out of nothing - omnibus ex nihilo ducendis sufficit unum.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned how Leibniz developed binary arithmetic - representing every number with only 0 and 1 - and why he saw deep significance in it. Explain how binary counting works and why it became the foundation of computing.
Leads to Alan Turing.
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